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A319458
Decimal expansion of the location of the minimum of the error term E(x) in exp(x) * x^(-x) * Pi^(-1/2) * Gamma(1 + x) = (8*x^3 + 4*x^2 + x + E(x))^(1/6) for x > 0. A319459 provides the corresponding value of E.
1
6, 7, 1, 5, 0, 3, 7, 6, 5, 7, 6, 8, 0, 2, 5, 3, 6, 0, 8, 6, 4, 8, 1, 2, 0, 5, 7, 5, 4, 0, 2, 3, 0, 0, 3, 4, 7, 3, 5, 0, 3, 2, 0, 7, 0, 1, 8, 0, 6, 0, 8, 1, 8, 3, 6, 5, 8, 3, 0, 8, 0, 4, 4, 8, 0, 3, 6, 3, 0, 8, 8, 5, 5, 9, 0, 6, 0, 1, 6, 4, 9, 5, 3, 9, 6, 8, 4, 1, 3, 5
OFFSET
0,1
COMMENTS
Ramanujan's question 754 in the Journal of the Indian Mathematical Society (VIII, 80) asked "Show that exp(x) * x^(-x) * Pi^(-1/2) * Gamma(1 + x) = (8*x^3 + 4*x^2 + x + E)^(1/6), where E lies between 1/100 and 1/30 for all positive values of x".
A numerical search provides an approximate minimum of E = 0.010045071877... (A319459) at x = 0.6715..., confirming Ramanujan's lower bound.
LINKS
B. C. Berndt, Y. S. Choi, S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56, DOI: 10.1090/conm/236 (see Q754, JIMS VIII).
B. C. Berndt, Y. S. Choi, S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q754, JIMS VIII).
EXAMPLE
0.6715037657680253608648120575402300347350320701806081836583...
CROSSREFS
Cf. A319459.
Sequence in context: A340926 A356562 A011483 * A296459 A354684 A308915
KEYWORD
nonn,cons
AUTHOR
Hugo Pfoertner, Sep 19 2018
STATUS
approved